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病的な関数

関数の不連続性について 

Def. set of discontinuity
Given a function f:R→R, define Df⊂R
to be the set of points where the function f fails to be continuouts.

Def. Fσ set
A set that can be written as the countable union of closed sets
is in the class Fσ

Th. 
Given f:R→R be an arbitrary function.
Then, Df is an Fσ set.

Cor. 
(with Baire's category theorem, the set of irrational points is not Fσ set.)
There is no function f that is continuous at every rational point
and discontinuous at every irrational point.
\neg {}^\exists f:\mathbb{R} \to \mathbb{R} \mbox{ s.t. } D_f = \mathbb{I}

Def. Dirichlet's function
g(x) := \begin{cases} 1 & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q}\end{cases}

a nowhere-continuous function on R
D_g = \mathbb{R}

Def. Modified Dirichlet's function
h(x) := \begin{cases} x & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q}\end{cases}

not continuous at every point x≠0
D_h = \mathbb{R} \setminus \{ 0 \}

Def. Thomae's function 1875
t(x) := \begin{cases} 1 & x=0 \\ \frac{1}{n} & x=\frac{m}{n} \in \mathbb{Q}^\times \mbox{ where }m,n \mbox{ is coprime and } n>0 \\ 0 & x \notin \mathbb{Q} \end{cases} 

t(x) fails to be continuous at any rational point.
whereas t(x) is continuous at every irrational point on R.
それぞれQに収束する点列・Iに収束する点列をとってみれば分かる。
D_t = \mathbb{Q}

\sin \frac{1}{x} は，区間(0,1]で連続かつ有界であるが、一様連続ではない

関数の微分可能性について 

Def. Weierstrass 1872
a class of continuous nowhere-differentiable function
f(x) := \sum_{n=0}^\infty a^n \cos (b^n x)
where the values of a and b are carefully chosen.

Def. Takagi
\sum_{n=0}^\infty \frac{1}{2^n} h(2^n x)
where h(x) := |x| \, \mod [-1,1]

Cor. of Baire's Category theorem
世の中の連続関数はだいたい Weierstrass 関数みたいに，全域で微分不能

Th. Lebesgue 1903
a continuous, monotone function would have to be differentiable at almost every point in its domain.
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最終更新：2010年11月10日 11:30
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